SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Avila-Bochi formula for certain products of 2× 2matrices
Joao Lopes Dias
Departamento de Matematica, ISEG and CemapreUniversidade Tecnica de Lisboa
23 Apr 2009
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Outline
1 SL(2,R)-cocycles
2 Herman’s theorem
3 Avila-Bochi improvement
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
SL(2, R)-cocycles
Let
(X,µ) compact probability space
f : X → X µ-preserving, ergodic
A : X → SL(2,R) measurable∫log ‖A‖ dµ < +∞
An(x) = A(fn−1(x)) . . . A(x)
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
KAN Iwasawa decomposition
A(x) = Rϕ(x)H(x)N(x) = Rϕ(x) T (x)
where
Rx =[cos(2πx) − sin(2πx)sin(2πx) cos(2πx)
], T (x) =
[c(x) b(x)0 c(x)−1
]and c(x) ≥ 1
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Remark
For A ∈ SL(2,R):
‖A‖ = sup‖v‖2=1 ‖Av‖2
‖A‖ =√ρ(ATA) =
√β +
√β2 − 1 where β = 1
2
∑ij A
2ij
‖A‖+ ‖A‖−1 =√
2(β + 1) = |c+ c−1 + ib|‖A‖ = ‖A−1‖ ≥ 1
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Outline
1 SL(2,R)-cocycles
2 Herman’s theorem
3 Avila-Bochi improvement
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
(Upper) (fiber) Lyapunov exponent
λ(A) = limn→+∞
1n
∫X
log ‖An(x)‖ dµ(x)
(generalizes to complex matrices)
Theorem (Herman)∫ 1
0λ(RθA) dθ ≥
∫X
log(‖A(x)‖+ ‖A(x)‖−1
2
)dµ(x)
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Example (Herman’s)
ϕ(x) = x, c(x) = c, b(x) = 0. Thus, RθA(x) = A(x+ θ) and
λ(A) = λ(RθA) ≥ log(c+ c−1
2
)
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Proof
Let
Sz = z
[z+z−1
2 − z−z−1
2iz−z−1
2iz+z−1
2
]
S0 = 12
[1 −ii 1
]Se2πit = e2πitRt
Fix x ∈ X. LetCz = Sze2πiϕ(x)T (x)
Ce2πiθ = e2πi(θ+ϕ(x))RθA(x)λ(RθA) = λ(Ce2πiθ)
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Lemma
The map C→ R+0 ,
z 7→ λ(Cz)
is subharmonic.
Therefore, ∫ 1
0λ(Ce2πiθ) dθ ≥ λ(C0) = λ(S0 T )
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Now,
LS0T (x)L−1 =
[c(x)+c(x)−1+ib(x)
2b(x)−ic(x)−1
20 0
]
=[α(x) β(x)
0 0
]where L =
[1 0−i 1
]‖(LS0TL
−1)n(x)‖ =∏n−1i=1 |α(f ix)|
∥∥∥[ α(x) β(x)0 0
]∥∥∥λ(S0T ) = λ(LS0TL
−1) ≥ lim1n
n−1∑i=0
log |α(f ix)|
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Using Birkhoff’s ergodic theorem∫ 1
0λ(RθA) dθ ≥ λ(LS0TL
−1)
≥∫X
log |α(x)| dµ(x)
=∫X
log|c(x) + c(x)−1 + ib(x)|
2dµ(x)
=∫X
log(‖A(x)‖+ ‖A(x)‖−1
2
)dµ(x)
End of the proof of theorem.
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Outline
1 SL(2,R)-cocycles
2 Herman’s theorem
3 Avila-Bochi improvement
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Avila-Bochi improvement
Theorem (Avila-Bochi-Herman)∫ 1
0λ(RθA) dθ =
∫X
log(‖A(x)‖+ ‖A(x)‖−1
2
)dµ(x)
Example (Herman’s revisited)
A(x) = Rx
[c 00 1c
],
λ(A) = λ(RθA) = log(c+ c−1
2
)
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Fix x ∈ X. Let
Bθ,n = (RθA)n(x) = RθA(fn−1x) . . . RθA(x)
N(A) = log(‖A‖+ ‖A‖−1
2
)
Theorem (Avila-Bochi formula)∫ 1
0log ρ(Bθ,n) dθ =
n−1∑i=0
N(A(f ix)) =∫ 1
0N(Bθ,n) dθ
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Proof of Theorem A-B-H
∫ 1
0λ(RθA) dθ = lim
1n
∫ 1
0log ‖Bθ,n‖ dθ (dominated conv thm)
= lim1n
∫ 1
0N(Bθ,n) dθ (| logA− logN(A)| ≤ log 2)
= lim1n
n−1∑i=0
N(A(f ix)) (A-B formula)
=∫XN(A(x)) dµ(x) (Birkhoff)
=∫X
log(‖A(x)‖+ ‖A(x)‖−1
2
)dµ(x)
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Proof of Avila-Bochi formula
Let for every z ∈ C
Sz = z
[z+z−1
2 − z−z−1
2iz−z−1
2iz+z−1
2
]Cz = SzA(x)
Se2πiθ = e2πiθRθ
(Ce2πiθ)n = e2πinθBθ,n
ρ(Bθ,n) = ρ((Ce2πiθ)n)det((C0)n) = det(S0)n = 0, thus 0 is e-value of (C0)n
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Proposition (A-B)
1 z 7→ log ρ((Cz)n) harmonic on D, continuous on D2 log ρ((C0)n) =
∑n−1i=0 N(A(f ix))
Remark
Herman proved that z 7→ λ(Cz) is subharmonic
First equality of Avila-Bochi formula
∫ 1
0log ρ(Bθ,n) dθ =
∫ 1
0log ρ((Ce2πiθ)n) dθ = log ρ((C0)n)
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Remark∫ 10 log ρ(Rθ′Bθ,n) dθ =
∫ 10 log ρ(Bθ,n) dθ∫ 1
0 log ρ(Rθ′Bθ,n) dθ′ = N(B0,n)
Second equality of Avila-Bochi formula
n−1∑i=0
N(A(f ix)) =∫ 1
0log ρ(Bθ,n) dθ
=∫ 1
0
∫ 1
0log ρ(Rθ′Bθ,n) dθ dθ′
=∫ 1
0N(Bθ,n) dθ
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Proof of Prop A-B
Want to show that (Cz)n = (SzA)n(x) and its e-valuesλ1(z), λ2(z), (holomorphic maps on C) verify:
z 7→ log ρ((Cz)n) = log maxi|λi(z)|
is harmonic on D, continuous on Dρ((C0)n) = maxi |λi(0)| =
∏n−1i=0
‖A(f ix)‖+‖A(f ix)‖−1
2
Finally
Enough to prove that |λ1(z)| 6= |λ2(z)|, z ∈ D, and computemaxi |λi(0)|
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Remark
|λ1| = |λ2| iff(trC)2
4 detC=
(λ1 + λ2)2
4λ1λ2∈ [0, 1]
det(Cz)n = z2n
Let the rational function Q : C∗ → C, holomorphic, be
Q(z) =trCz2zn
deg(Q) ≤ 2n and deg(Q′) ≤ 2n
Thus|λ1(z)| = |λ2(z)| iff Q(z) ∈ [−1, 1]
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
So
Want to show that Q(D) ∩ [−1, 1] = ∅
Q(e2πiθ) = 12 tr(Bθ,n) ∈ [−1, 1] iff Bθ,n is non-hyperbolic
(| tr | ≤ 2)
Q(∂D) ⊂ R
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Lemma (A-B)
If ∂D ∩Q−1([−1, 1]) has at least 2n connected components, then
Q(D) ∩ [−1, 1] = ∅
Proof.
deg(Q′) ≤ 2n hence ≤ 2n zeros of Q′
Each component of the complement in ∂D has a zero of Q′
Thus, there are no zeros of Q′ in Q−1([−1, 1])So, each component is diffeo to [−1, 1]
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SL(2,R)-cocyclesHerman’s theorem
Avila-Bochi improvement
Lemma (A-B)
∂D ∩Q−1([−1, 1]) has at least 2n connected components
Proof.
Uses a topological argument...
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